Paper LXXIII: Quantum Stochasticity from Sub-Planck Fano Node Superposition: Radioactive Decay Rates as Planck-Scale Poisson Processes, the Born Rule from Fano Overlap Integrals, and the Resolution of the Hidden Variable Problem

The stochastic nature of quantum mechanical decay the irreducible un- predictability of which specic particle decays and precisely when has no mechanistic explanation in the Standard Model, where it is treated as a fundamental axiom. We show that the brane-bulk octonionic framework derives this stochasticity from a concrete sub-Planck mechanism: the superposition of Fano lattice node occupations at energy scales well below the Planck mass. At the energy of a neutron (mn ≈0. 94 GeV), the d quark focal zone is delocalized across the SU (2) L isospin doublet nodes e1, e2 of the Fano lattice with a frame orientation uncertainty δφ4 (mn) = 11. 0 ℓP (Paper LXII). This superposition uctuates on the Planck timescale tPl ≈5. 4 × 10−44 s. (1) Decay occurs when the uctuating Fano node superposition achieves sucient over- lap with the W −reconguration channel. At each Planck timestep, the d quark node superposition |ψ (t) ⟩= cos θ (t) |e2⟩+ sin θ (t) |e1⟩uctuates. When | sin θ (t) |2 ex- ceeds the sub-threshold tunneling amplitude GF m2 n ≈10−5, the transition d →u + W − occurs. (2) The exponential decay law is the classical Poisson statistics of a Planck-scale process. Each Planck timestep is an independent trial with probability p = ΓntPl/ℏ≈6. 1×10−47. The decay-time distribution is therefore geometric, which for small p reduces to an exponential with mean lifetime τ = ℏ/Γn exactly the quantum result. The exponential decay law is not a quantum mystery; it is the Poisson statistics of the underlying Planck-scale stochastic process. (3) The Born rule is the squared Fano overlap integral. The transition probability is |⟨W −channel|d quark state⟩|2 the squared overlap between the delocalized Fano node superposition and the transition state. This is the Born rule, derived from the Fano geometry rather than postulated. (4) The stochasticity is epistemic from the brane, deterministic from the bulk. The specic Fano node trajectory of the d quark is a denite path in H3 (the hyperbolic 3-space of Lorentzian frame orientations, Paper LXV), but this trajectory is inaccessible to any observer conned to the brane. The hidden variable is the sub- Planck bulk trajectory genuinely hidden by the brane horizon, not merely practically inaccessible. This resolves Bell's theorem: non-local bulk correlations (Paper LXV) al- low Bell inequality violation without superluminal signalling. Three new predictions follow (Predictions 158160). 355 Brane-Bulk Octonionic Series B. D. Jagadeesan MD Chapter 5: . . . Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10. 5281/zenodo. 19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M. D. , University of Minnesota. ORCID: 0000-0002-1143-941X.